Nonrecursive canonical basis computations for low rank Kashiwara crystals of type A

Quantum cluster algebras of type A and the dual canonical basis

The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the structure of a quantum cluster algebra of type A_n. The quantum cluster algebra is a deformation of the ordinary cluster algebra Geiss-Leclerc-Schroeer attached to w using the representation theory of the preprojective algebra. Furthermore, we prove that the quantum cluster variables are, up to a power of v, elements in the dual of Lusztig's canonical basis und…

The Maslov index as a quadratic space

Kashiwara defined the Maslov index (associated to a collection of Lagrangian subspaces of a symplectic vector space over a field F) as a class in the Witt group W(F) of quadratic forms. We construct a canonical quadratic vector space in this class and show how to understand the basic properties of the Maslov index without passing to W(F)--that is, more or less, how to upgrade Kashiwara's equalities in W(F) to canonical isomorphisms between quadratic spaces. We also show how our canonical quadratic form occurs naturally in the context of the Weil representation. The quadratic space is defined u…

The canonical basis and the quantum Frobenius morphism

The first goal of this paper is to study the amount of compatibility between two important constructions in the theory of quantized enveloping algebras, namely the canonical basis and the quantum Frobenius morphism. The second goal is to study orders with which the Kashiwara crystal B(infinity) of a symmetrizable Kac-Moody algebra can be endowed; these orders are defined so that the transition matrices between bases naturally indexed by B(infinity) are lower triangular.

Periodic modules and quantum groups

We prove that the elements $A_\leq$ defined by Lusztig in a completion of the periodic module actually live in the periodic module, in the type A case. In order to prove this, we compare, using Schur duality, these elements with the Kashiwara canonical basis of an integrable module.

Combinatorics of canonical bases revisited: String data in type $A$

We give a formula for the crystal structure on the integer points of the string polytopes and the $*$-crystal structure on the integer points of the string cones of type $A$ for arbitrary reduced words. As a byproduct we obtain defining inequalities for Nakashima-Zelevinsky string polytopes. Furthermore, we give an explicit description of the Kashiwara $*$-involution on string data for a special choice of reduced word.

典范基的分拆

Partitions of canonical bases

该研究证明了Lusztig与Kashiwara构造的量子群典范基的各类分拆重合，据此得出开Richardson簇对应子集与Schubert胞腔对应子集的交集关系，并证明A型中开Richardson簇的权在对应Bruhat区间多面体中饱和。

Tensor product multiplicities, canonical bases and totally positive varieties

We obtain a family of explicit "polyhedral" combinatorial expressions for multiplicities in the tensor product of two simple finite-dimensional modules over a complex semisimple Lie algebra. Here "polyhedral" means that the multiplicity in question is expressed as the number of lattice points in some convex polytope. Our answers use a new combinatorial concept of $\ii$-trails which resemble Littelmann's paths but seem to be more tractable. We also study combinatorial structure of Lusztig's canonical bases or, equivalently of Kashiwara's global bases. Although Lusztig's and Kashiwara's approach…

Positive representations, multiplier Hopf algebra, and continuous canonical basis

We introduce the language of multiplier Hopf algebra in the context of positive representations of split real quantum groups, and discuss its applications with a continuous version of Lusztig-Kashiwara's canonical basis, which may provide a key to prove the closure of the positive representations under tensor products, and harmonic analysis of quantized algebra of functions in the sense of locally compact quantum groups.

Crystals via the affine Grassmannian

Let $G$ be a connected reductive group over $\CC$ and let $G^{\vee}$ be the Langlands dual group. Crystals for $G^{\vee}$ were introduced by Kashiwara as certain ``combinatorial skeletons'' of finite-dimensional representations of $G^{\vee}$. For every dominant integral weight of $G^{\vee}$ Kashiwara constructed a canonical crystal. Other (independent) constructions of those crystals were given by Lusztig and Littelmann. It was also shown by Kashiwara and Joseph that the above family of crystals is unique if certain reasonable conditions are imposed. The purpose of this paper is to give anothe…

注记：从对偶典范基到量子丛代数的三角基

Notes: from dual canonical bases to triangular bases of quantum cluster algebras

本文证明任意可对称化 Kac–Moody 代数对应量子幺幂子群的对偶典范基即量子丛代数的公共三角基，并据此说明其包含所有量子丛单项式。

The correspondence between the canonical and semicanonical bases

Given any symmetric Cartan datum, Lusztig has provided a pair of key lemmas to construct the perverse sheaves over the corresponding quiver and the functions of irreducible components over the corresponding preprojective algebra respectively. In the present article, we prove that these two inductive algorithms of Lusztig coincide. Consequently we can define two colored graphs and prove that they are isomorhic. This result finishes the statement that Lusztig's functions of irreducible components are basis of the enveloping algebra and deduces the crystal structure (in the sense of Kashiwara-Sai…

Lectures on canonical and crystal bases of Hall algebras

These are the notes for a series of lectures given on the theory of canonical and crystal bases for Hall algebras (for a summer school in Grenoble in 2008). It may be viewed as a follow-up to arXiv:math/0611617. It covers the construction, due to Lusztig, of the canonical bases for the Hall algebra of a quiver Q in terms of a certain category of perverse sheaves over the moduli space of representations of Q. It also contains an exposition of Kashiwara and Saito's geometric construction of the crystal graph in terms of irreducible components of Lusztig's lagrangian in the cotangent bundle to th…

Affine sl_p controls the representation theory of the symmetric group and related Hecke algebras

In this paper we prove theorems that describe how the representation theory of the affine Hecke algebra of type A and of related algebras such as the group algebra of the symmetric group are controlled by integrable highest weight representations of the characteristic zero affine Lie algebra \hat{sl}_l. In particular we parameterise the representations of these algebras by the nodes of the crystal graph, and give various Hecke theoretic descriptions of the edges. As a consequence we find for each prime p a basis of the integrable representations of \hat{sl}_l which shares many of the remarkabl…

Combinatorics of canonical bases revisited: Type A

We initiate a new approach to the study of the combinatorics of several parametrizations of canonical bases. In this work we deal with Lie algebras of type $A$. Using geometric objects called Rhombic tilings we derive a "crossing formula" to compute the actions of the crystal operators on Lusztig data for an arbitrary reduced word of the longest Weyl group element. We provide the following three applications of this result. Using the tropical Chamber Ansatz of Berenstein-Fomin-Zelevinsky we prove an enhanced version of the Anderson-Mirković conjecture for the crystal structure on MV polytopes.…

微局部Arthur包作为稳定分布的拟典范基

Microlocal Arthur packets as a quasi-canonical basis of stable distributions

本文利用Kashiwara局部指标定理的线性方程组，结合Arthur轨道几何刚性与傅里叶对称性，重构微局部Arthur包，给出一般连通约化p进群的替代定义，证明分裂SO_{2n+1}的相关包一致性，计算例外群的幂幺Arthur包并证明相关猜想、给出反例。

Temperley-Lieb 晶体

Temperley-Lieb Crystals

本文针对 Temperley-Lieb 子式推广 Littlewood-Richardson 法则，引入洗牌 tableau，并利用 Stembridge 公理证明其构成 A 型 Kashiwara 晶体，从而给出相关 Schur 系数的组合解释。

次正则仿射胞与D型-1级顶点代数

Subregular affine cells and the level $-1$ vertex algebra of type $D$

该研究证明了Shan-Yan-Zhao仿射左胞猜想对$\ell\ge5$的$L_{-1}(D_\ell)$成立，明确真空块单对象数量，推导了Grothendieck群的特殊化及统一特征标公式。